Characteristic box dimension of unit-time map near nilpotent singularity of planar vector field and applications

TL;DR

This paper explores how the box dimension of the unit-time map can analyze nilpotent singularities in planar vector fields, revealing multiplicity, cyclicity bounds, and applications to bifurcations.

Contribution

It introduces the concept of characteristic box dimension of the unit-time map and connects it to the multiplicity and bifurcation properties of nilpotent singularities.

Findings

Box dimension relates to singularity multiplicity.

Box dimension of Poincaré map bounds cyclicity.

Application to Bogdanov-Takens bifurcation.

Abstract

This article shows how box dimension of the unit-time map can be used in studying the multiplicity of nilpotent singularities of planar vector fields. Using unit-time map on the characteristic curve of nilpotent singularity we define characteristic map and characteristic box dimension of the unit-time map. We study connection between the box dimension of discrete orbits generated by the unit-time map of planar vector fields on the characteristic or invariant curves, and the multiplicity of singularities. Nilpotent singularities which are studied are nilpotent node, focus and cusp. Also, we show how box dimension of the Poincar\' e map near the nilpotent focus on the characteristic curve reveals the upper bound for cyclicity. Moreover, we study the characteristic box dimension of nilpotent cusp at infinity which is connected to the order of cusp. At the end, we applied the results of box dimension of unit-time map to the Bogdanov-Takens bifurcation.